Is the Distance Compression Effect Overstated? Some Theory and Experimentation

Abstract

Previous work in the document clustering literature has shown that the Minkowski-p distance metrics are unsuitable for clustering very high dimensional document data. This unsuitability is put down to the effect of “compression” of the distances created using the Minkowski-p metrics on high dimensional data. Previous experimental work on distance compression has generally used the performance of clustering algorithms on distances created by the different distance metrics as a proxy for the quality of the distance representations created by those metrics. In order to separate out the effects of distances from the performance of the clustering algorithms we tested the homogeneity of the latent classes with respect to item neighborhoods rather than testing the homogeneity of clustering solutions with respect to latent classes. We show the theoretical relationships between the cosine, correlation, and Euclidean metrics. We posit that some of the performance differential between the cosine and correlation metrics and the Minkowski-p metrics is due to the inbuilt normalization of the cosine and correlation metrics. The normalization effect decreases with increasing dimensionality and the distance compression effect increases with increasing dimensionality. For document datasets with dimensionality up to 20,000, the normalization effect dominates the distance compression effect. We propose a methodology for measuring the relative normalization and distance compression effects.